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Properties

Label 1-1183-1183.1101-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.635 - 0.771i$
Analytic cond. $5.49382$
Root an. cond. $5.49382$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.200 − 0.979i)2-s + (−0.354 − 0.935i)3-s + (−0.919 + 0.391i)4-s + (0.692 + 0.721i)5-s + (−0.845 + 0.534i)6-s + (0.568 + 0.822i)8-s + (−0.748 + 0.663i)9-s + (0.568 − 0.822i)10-s + (−0.748 − 0.663i)11-s + (0.692 + 0.721i)12-s + (0.428 − 0.903i)15-s + (0.692 − 0.721i)16-s + (0.428 − 0.903i)17-s + (0.799 + 0.600i)18-s + 19-s + (−0.919 − 0.391i)20-s + ⋯
L(s)  = 1  + (−0.200 − 0.979i)2-s + (−0.354 − 0.935i)3-s + (−0.919 + 0.391i)4-s + (0.692 + 0.721i)5-s + (−0.845 + 0.534i)6-s + (0.568 + 0.822i)8-s + (−0.748 + 0.663i)9-s + (0.568 − 0.822i)10-s + (−0.748 − 0.663i)11-s + (0.692 + 0.721i)12-s + (0.428 − 0.903i)15-s + (0.692 − 0.721i)16-s + (0.428 − 0.903i)17-s + (0.799 + 0.600i)18-s + 19-s + (−0.919 − 0.391i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.635 - 0.771i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.635 - 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9479003483 - 0.4471688539i\)
\(L(\frac12)\) \(\approx\) \(0.9479003483 - 0.4471688539i\)
\(L(1)\) \(\approx\) \(0.7362419162 - 0.4192424021i\)
\(L(1)\) \(\approx\) \(0.7362419162 - 0.4192424021i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.200 - 0.979i)T \)
3 \( 1 + (-0.354 - 0.935i)T \)
5 \( 1 + (0.692 + 0.721i)T \)
11 \( 1 + (-0.748 - 0.663i)T \)
17 \( 1 + (0.428 - 0.903i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.948 + 0.316i)T \)
31 \( 1 + (-0.845 + 0.534i)T \)
37 \( 1 + (-0.845 + 0.534i)T \)
41 \( 1 + (-0.632 + 0.774i)T \)
43 \( 1 + (-0.0402 + 0.999i)T \)
47 \( 1 + (0.799 - 0.600i)T \)
53 \( 1 + (-0.996 + 0.0804i)T \)
59 \( 1 + (0.692 + 0.721i)T \)
61 \( 1 + (0.568 - 0.822i)T \)
67 \( 1 + (0.120 + 0.992i)T \)
71 \( 1 + (0.987 + 0.160i)T \)
73 \( 1 + (0.948 - 0.316i)T \)
79 \( 1 + (0.799 - 0.600i)T \)
83 \( 1 + (-0.354 + 0.935i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.692 - 0.721i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.409734580419144730338424972863, −20.656416458699843618982364031461, −19.984292209646077976121718797847, −18.72807046106677609775472259430, −17.88533960989719190372406477619, −17.31734346134796843151582047797, −16.70354524018736094324932850947, −15.89718933080205917565750278275, −15.46142503421650627137409654364, −14.42225568349996104276694253629, −13.868440728551553807440425885125, −12.75919319443728295497076270541, −12.16808292519303873026111956475, −10.64619321445330927282910251285, −10.11174587160858144866774344007, −9.452369612753037732127545315523, −8.64753088467253032029881063831, −7.89407433769004346391560435784, −6.72033725965252806927042455616, −5.76319810810398575159760678629, −5.27413923670186876455830177519, −4.525313107012304142820479261114, −3.61525653400441835329595872822, −2.05041810175316062250615387777, −0.60942998343141216906089242047, 0.944129477840183085560179317468, 1.85742151758751286639690357009, 2.84786396752208430248605976122, 3.33319455403079299356349473657, 5.12616849824315827397760636370, 5.52771847398408397636442654728, 6.73168169678121489606487752247, 7.58322112895475170852605580609, 8.3440395583796752026901029667, 9.45379672202297149670633156264, 10.194982443326439965836643542272, 11.02474757621321775516050202915, 11.61781280731674998741197359637, 12.41436716499682658733894178513, 13.38103279117010544275319858785, 13.80596581678272481692848615088, 14.38981132241063568368683497771, 15.90289599496819554393356889333, 16.791344637837618721435562792709, 17.73637691755392545491382865263, 18.18907453318845202502668645489, 18.656258178214743014836086145454, 19.45295761963058769603564814056, 20.253139473919761241005735204696, 21.14828012019304831213629236810

Graph of the $Z$-function along the critical line